A load of mass `m` is attached to the end of a string wound on a cylinder of mass `M` and radius `R`. The string passes round a pulley.
a. Find the acceleration of the cylinder `M` when it rolls.
b. If the coefficient of friction is `mu`. Find the accelerative of the cylinder when it slips and rolls.
c. Find the minimum value of coefficient of friction for which the cylinder rolls always

Explanation

The cylinder will roll without slipping so long as the friction force is suficiently large and it never reaches the limiting value. The force acting on the cylinder are a. its weight `Mg`, b. normal reaction `N`, c. tension `T` to the right, d `f_("fr")` to the right. let a be the acceleration of the falling mass ,m.
Then `a_(CM)=a/2`

`mg-T=ma`
`impliesT=mg-ma`
For the cylinder
`T+f_("fr")=(Ma)/2`
and `tau=TR-f_("fr")R=(1/2MR^(2))alpha`
`T-f_("fr")=1/2MRalpha=1/2MR(a_(CM)/R)=1/2Ma_(CM)=1/4 Ma`
Solving `f_("fr")=1/8 Ma`
`T=3/8 Ma and a=(mg)/(m+3/8M)`
There will be no slip if
`f_("fr")lemumgimpliesmuge(Ma)/(8mg)`
or `mugtM/(8m=3M)`
Rolling `+`slipping: Now `a_(CM)!=alphaR`, rather acceleration of the highest point of the cylinder `=a_(CM)+alphaR=` acceleration of `m`.
and `tau=TR-f_(lim)R=(1/2MR^(2))alpha`
`T-muMg=1/2MRalpha`
For load `mg-t=ma`
Solving for `alpha`
`a=(3mg-muMg)/(M+3m)`